This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.
\begin{gather*}
\Pr(a \leq X \leq b) = \int_a^b f(x; \mu, \sigma^2)\ dx.
\end{gather*}
Unfortunately, \(e^{-x^2}\) has no elementary antiderivative. But we know \(\int_a^b f(x)\ dx\) represents the area under the graph \(y = f(x; \mu, \sigma^2)\text{,}\) and this area can be approximated to arbitrary precision.
Suppose \(X_1, \dotsc, X_n\) are independent and identically distributed (or i.i.d.) with finite expected value \(\mu\) and finite variance \(\sigma^2\text{.}\) Define:
We’ll omit a proof of Theorem 117, but we can at least check that the expected values and variances of \(S_n, A_n\) are correct. Given \(\E(X_i) = \mu, \Var(X_i) = \sigma^2\text{:}\)
Then \(Z \sim \Norm(0, 1)\text{.}\) Values of \(Z\) are called \(z\)-scores, sometimes indicated by an asterisk. (I.e., if \(a\) is a value of \(S\text{,}\) then \(a^*\) is the corresponding \(z\)-score.) Now:
Now, we still can’t compute the value of the integral. However, there’s a huge benefit to translating to the standard normal distribution, regardless of which normal distribution we used to approximate \(S\text{.}\) We can approximate integrals like \(\int_a^b \phi(x)\ dx\) to abritrary precision and record the results in a table. Then, we can look up the values when needed. We don’t need to redo our approximations for different normal distributions, we just standardize whatever normal distribution we come across.
